Collaborative Research: Multilevel Methods for Optimal Control of Partial Differential Equations and Optimization-Based Domain Decomposition

微分方程最优控制（PDECO）在越来越多的实际应用中发挥着重要作用，从油藏建模到天气预报和飞机机翼的优化形状设计。虽然传统的PDECO使用确定性模型，但该项目针对的PDECO的微分方程也包含不确定性，例如地面成分的不规则波动或湍流风速。最终目标是显著提高这类最优控制问题的解质量和计算时间。该项目产生的新算法将影响地球物理、天气建模等领域的优化问题。这些问题是普遍的，解决技术的进步也将使其他科学受益。开源软件将被创建并与社区共享。四名研究生将从该项目中受益。将特别注意从代表性不足的群体中招收学生。该项目主要针对两类潜在的大规模PDECO问题：受随机偏微分方程（PDEs）约束的PDECO和受非局部偏微分方程约束的PDECO，开发鲁棒的、可扩展的多级求解器。一个额外的推动力是为前向pde模型本身开发支持基于优化的域分解（另一种PDECO）的多级求解器。已知多级/多网格解算器对于许多类型的正演模型是最优的。然而，它们在解决PDECO问题上的应用仍处于起步阶段。幼稚地应用多层方法来解决这类优化问题可能导致依赖于分辨率（网格依赖）和问题的其他参数，如随机维数或子域的数量。此外，由于每次迭代至少涉及一个PDE解决方案，因此解决此类优化问题的成本对于大规模的高分辨率问题来说可能是令人望而生畏的，特别是对于那些比传统的确定性问题要昂贵得多的问题。本项目开发的算法旨在建立新的效率和鲁棒性标准。新颖的数学工具将进一步推进数值分析和优化的知识。新的专题课程将基于项目中产生的研究而开发，笔记将与社区共享。研究结果将通过技术研究论文和在国家和国际会议上的演讲积极传播。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Optimal control of differential equations (PDECO) plays an important role in an ever increasing number of real-life applications ranging from petroleum reservoir modeling to weather prediction and the optimal shape design of airplane wings. While traditional PDECO uses deterministic models, this project targets PDECO where the differential equations also include uncertainties, such as irregular fluctuations in the ground composition, or turbulent wind speeds. The ultimate aim is to dramatically improve the solution quality and the computing time of such optimal control problems. The novel algorithms resulted from this project will impact optimization problems arising in geophysics, weather modeling etc. These problems are generic and advances in solution techniques will also benefit other sciences. Open source software will be created and shared with the community. Four graduate students will benefit from the project. Special attention will be given to recruit students from underrepresented groups. The project is focused on developing robust, scalable multilevel solvers for mainly two classes of potentially large-scale PDECO problems: PDECOs constrained by stochastic partial differential equations (PDEs) and by nonlocal PDEs. An additional thrust is to develop multilevel solvers in support of optimization-based domain decomposition - another kind of PDECO - for the forward PDE-models themselves. Multilevel/multigrid solvers are known to be optimal for many classes of forward models. However, their application to solve PDECO problems is still in its infancy. A naive application of multilevel methods to solve such optimization problems can lead to dependence on resolution (mesh-dependence) and on other parameters of the problem such as the stochastic dimension or the number of subdomains. In addition, since each iterate involves at least one PDE solve, the cost of solving such optimization problems can be prohibitive for large-scale, high-resolution problems, especially for problems that are significantly more expensive than the traditional, deterministic ones. The algorithms developed in this project aim to set new standards of efficiency and robustness. Novel mathematical tools will further advance the knowledge in numerical analysis and optimization. New special topics courses will be developed based on the research generated in the project and the notes will be shared with the community. The results of the research will be actively disseminated via technical research papers and talks at national and international conferences.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.